$q,t$-定义$(\C^2)^n$对角轨迹的radical理想的Catalan数和生成元
$q,t$-Catalan numbers and generators for the radical ideal defining the diagonal locus of $(\C^2)^n$
摘要：
令$I$是由两组$n$变量中的交替多项式生成的理想。Haiman证明了$q,t$-Catalan数是分次向量空间$M(=\bigoplus_{d_1,d_2}M_{d_1,d_2})$的生成元的极小集所张成的Hilbert级数。本文给出了$\text{dim}M_{d_1, d_2}$的一个简单的关于划分数的上界，并找到了所有的二度$(d_1,d_2)$使得$\dim M_{d_1, d_2}$达到上界.对于这样的双度，我们也发现明确的基地$M_{d_1, d_2}$。主要思想是定义和研究从$M$到多项式环$\C[ρ_1, ρ_2,...]$的非平凡线性映射。
英文摘要：
Let $I$ be the ideal generated by alternating polynomials in two sets of $n$ variables. Haiman proved that the $q,t$-Catalan number is the Hilbert series of the graded vector space $M(=\bigoplus_{d_1,d_2}M_{d_1,d_2})$ spanned by a minimal set of generators for $I$. In this paper we give simple upper bounds on $\text{dim}M_{d_1, d_2}$ in terms of partition numbers, and find all bi-degrees $(d_1,d_2)$ such that $\dim M_{d_1, d_2}$ achieve the upper bounds. For such bi-degrees, we also find explicit bases for $M_{d_1, d_2}$. The main idea is to define and study a nontrivial linear map from $M$ to a polynomial ring $\C[ρ_1, ρ_2,...]$.
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